This paper introduces the concept of the transient hyperbolic exponent b. The transient hyperbolic exponent may be used to derive values for b for use in Arp's equations for a variety of drive mechanisms. For a coalbed methane reservoir, b is shown to be a function of the ratio of the average reservoir pressure to the Langmuir pressure, with b approaching 1.5 at pressures much higher than pL, 1.0 when the pressure is equal to pL, and approaching 0.5 at pressures much lower than pL. For wells producing from finite, transient dual porosity reservoirs, the b value may be as high as 2, during a flow period characterized by steady state flow in the natural fracture system, and transient linear flow in the matrix. A coalbed methane well and two Antrim Shale wells are presented as examples.
Arp's hyperbolic decline curve equation has been used for in the petroleum industry for almost 60 years to forecast future production by extrapolation of existing production trends. The hyperbolic exponent b appearing in the Arps equation has been a source of much controversy through the years, with many authors condemning the use of b values greater than 1, and others freely using any b value that fit the data.
Fetkovich combined the Arps equations with the constant terminal pressure solution to the diffusivity equation to produce a set of type curves for production data analysis. The resulting type curves have proven to be one of the most useful tools in analysis of production data.
Fetkovich et al. tabulated values for the hyperbolic exponent b as a function of reservoir drive mechanism. The authors state categorically that "The harmonic decline exponent, b=1, cannot be obtained. In fact, no other investigators have been able to derive an exponent b greater than 0.5 for any reasonable single-layer, homogeneous reservoir system or drive mechanism." However, they did not consider the case of coalbed methane.
Prats, Hazebroek, and Strickler showed that a hydraulically fractured well, producing at constant pressure from an infinite-acting reservoir, would exhibit a period of linear flow, where the rate was inversely proportional to the square root of time. Maley showed that this period of linear flow would, if matched with Arp's equation, result in a b value of 2. In low permeability formations with long hydraulic fractures, this linear flow period may last quite some time. However, once the boundaries of the reservoir are felt, the b value drops to a number less than 1.